Physica D: Nonlinear Phenomena [SJR: 0.976] [H-I: 83] [5 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 0167-2789 Published by Elsevier [2589 journals] |
- Kadomtsev–Petviashvili II equation: Structure of asymptotic soliton
webs- Abstract: Publication date: Available online 27 February 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Shai Horowitz , Yair Zarmi
A wealth of observations, recently supported by rigorous analysis, indicate that, asymptotically in time, most multi-soliton solutions of the Kadomtsev–Petviashvili II equation self-organize in webs comprised of solitons and soliton-junctions. Junctions are connected in pairs, each pair—by a single soliton. The webs expand in time. As distances between junctions grow, the memory of the structure of junctions in a connected pair ceases to affect the structure of either junction. As a result, every junction propagates at a constant velocity, which is determined by the wave numbers that go into its construction. One immediate consequence of this characteristic is that asymptotic webs preserve their morphology as they expand in time. Another consequence, based on simple geometric considerations, explains why, except in special cases, only 3-junctions (“ Y -shaped”, involving three wave numbers) and 4-junctions (“ X -shaped”, involving four wave numbers) can partake in the construction of an asymptotic soliton web.
PubDate: 2015-03-01T08:36:44Z
- Abstract: Publication date: Available online 27 February 2015
- Considerations on conserved quantities and boundary conditions of the
2+1-dimensional nonlinear Schrödinger equation- Abstract: Publication date: Available online 26 February 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Javier Villarroel , Julia Prada
In this study, we consider a natural integrable generalization of the defocusing cubic nonlinear Schrödinger equation to two dimensions and we classify the admissible boundary conditions. In particular, we determine whether the classical physical observables are conserved: mass, momentum, and Hamiltonian. We find that this is the case when a certain integral (the mass constraint) vanishes. The vanishing of the mass constraint, and thus the existence of conserved quantities, is contingent on the boundary conditions adopted. In particular, under decaying boundary conditions, the Hamiltonian is not necessarily conserved.
PubDate: 2015-03-01T08:36:44Z
- Abstract: Publication date: Available online 26 February 2015
- Modeling disease transmission near eradication: An equation free approach
- Abstract: Publication date: 1 January 2015
Source:Physica D: Nonlinear Phenomena, Volume 290
Author(s): Matthew O. Williams , Joshua L. Proctor , J. Nathan Kutz
Although disease transmission in the near eradication regime is inherently stochastic, deterministic quantities such as the probability of eradication are of interest to policy makers and researchers. Rather than running large ensembles of discrete stochastic simulations over long intervals in time to compute these deterministic quantities, we create a data-driven and deterministic “coarse” model for them using the Equation Free (EF) framework. In lieu of deriving an explicit coarse model, the EF framework approximates any needed information, such as coarse time derivatives, by running short computational experiments. However, the choice of the coarse variables (i.e., the state of the coarse system) is critical if the resulting model is to be accurate. In this manuscript, we propose a set of coarse variables that result in an accurate model in the endemic and near eradication regimes, and demonstrate this on a compartmental model representing the spread of Poliomyelitis. When combined with adaptive time-stepping coarse projective integrators, this approach can yield over a factor of two speedup compared to direct simulation, and due to its lower dimensionality, could be beneficial when conducting systems level tasks such as designing eradication or monitoring campaigns.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 1 January 2015
- Tracking pattern evolution through extended center manifold reduction and
singular perturbations- Abstract: Publication date: Available online 7 February 2015
Source:Physica D: Nonlinear Phenomena
Author(s): L. Sewalt , A. Doelman , H.G.E. Meijer , V. Rottschäfer , A. Zagaris
In this paper we develop an extended center manifold reduction method: a methodology to analyze the formation and bifurcations of small-amplitude patterns in certain classes of multi-component, singularly perturbed systems of partial differential equations. We specifically consider systems with a spatially homogeneous state whose stability spectrum partitions into eigenvalue groups with distinct asymptotic properties. One group of successive eigenvalues in the bifurcating group are widely interspaced, while the eigenvalues in the other are stable and cluster asymptotically close to the origin along the stable semi-axis. The classical center manifold reduction provides a rigorous framework to analyze destabilizations of the trivial state, as long as there is a spectral gap of sufficient width. When the bifurcating eigenvalue becomes commensurate to the stable eigenvalues clustering close to the origin, the center manifold reduction breaks down. Moreover, it cannot capture subsequent bifurcations of the bifurcating pattern. Through our methodology, we formally derive expressions for low-dimensional manifolds exponentially attracting the full flow for parameter combinations that go beyond those allowed for the (classical) center manifold reduction, i.e. to cases in which the spectral gap condition no longer can be satisfied. Our method also includes an explicit description of the flow on these manifolds and thus provides an analytical tool to study subsequent bifurcations. Our analysis centers around primary bifurcations of transcritical type–that can be either of co-dimension 1 or 2–in two- and three-component PDE systems. We employ our method to study bifurcation scenarios of small-amplitude patterns and the possible appearance of low-dimensional spatio-temporal chaos. We also exemplify our analysis by a number of characteristic reaction-diffusion systems with disparate diffusivities.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: Available online 7 February 2015
- On extreme events for non-spatial and spatial branching Brownian motions
- Abstract: Publication date: Available online 9 February 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Jean Avan , Nicolas Grosjean , Thierry Huillet
We study the impact of having a non-spatial branching mechanism with infinite variance on some parameters (height, width and first hitting time) of an underlying Bienaymé–Galton–Watson branching process. Aiming at providing a comparative study of the spread of an epidemics whose dynamics is given by the modulus of a branching Brownian motion (BBM) we then consider spatial branching processes in dimension d , not necessarily integer. The underlying branching mechanism is either a binary branching model or one presenting infinite variance. In particular we evaluate the chance p ( x ) of being hit if the epidemics started away at distance x . We compute the large x tail probabilities of this event, both when the branching mechanism is regular and when it exhibits very large fluctuations.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: Available online 9 February 2015
- Groove growth by surface subdiffusion
- Abstract: Publication date: Available online 10 February 2015
Source:Physica D: Nonlinear Phenomena
Author(s): M. Abu Hamed , A.A. Nepomnyashchy
The investigation of the grain-boundary groove growth by normal surface diffusion was first done by Mullins. However, the diffusion on a solid surface is often anomalous. Recently, the groove growth in the case of surface superdiffusion has been analyzed. In the present paper, the problem of the groove growth is solved in the case of the surface subdiffusion. An exact self-similar solution is obtained and represented in terms of the Fox H-function. Basic properties of the solution are described.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: Available online 10 February 2015
- Blended particle methods with adaptive subspaces for filtering turbulent
dynamical systems- Abstract: Publication date: Available online 11 February 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Di Qi , Andrew J. Majda
It is a major challenge throughout science and engineering to improve uncertain model predictions by utilizing noisy data sets from nature. Hybrid methods combining the advantages of traditional particle filters and the Kalman filter offer a promising direction for filtering or data assimilation in high dimensional turbulent dynamical systems. In this paper, blended particle filtering methods that exploit the physical structure of turbulent dynamical systems are developed. Non-Gaussian features of the dynamical system are captured adaptively in an evolving-in-time low dimensional subspace through particle methods, while at the same time statistics in the remaining portion of the phase space are amended by conditional Gaussian mixtures interacting with the particles. The importance of both using the adaptively evolving subspace and introducing conditional Gaussian statistics in the orthogonal part is illustrated here by simple examples. For practical implementation of the algorithms, finding the most probable distributions that characterize the statistics in the phase space as well as effective resampling strategies is discussed to handle realizability and stability issues. To test the performance of the blended algorithms, the forty dimensional Lorenz 96 system is utilized with a five dimensional subspace to run particles. The filters are tested extensively in various turbulent regimes with distinct statistics and with changing observation time frequency and both dense and sparse spatial observations. In real applications perfect dynamical models are always inaccessible considering the complexities in both modeling and computation of high dimensional turbulent system. The effects of model errors from imperfect modeling of the systems are also checked for these methods. The blended methods show uniformly high skill in both capturing non-Gaussian statistics and achieving accurate filtering results in various dynamical regimes with and without model errors.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: Available online 11 February 2015
- Rogue wave spectra of the Sasa–Satsuma equation
- Abstract: Publication date: 15 February 2015
Source:Physica D: Nonlinear Phenomena, Volume 294
Author(s): N. Akhmediev , J.M. Soto-Crespo , N. Devine , N.P. Hoffmann
We analyze the rogue wave spectra of the Sasa–Satsuma equation and their appearance in the spectra of chaotic wave fields produced through modulation instability. Chaotic wave fields occasionally produce high peaks that result in a wide triangular spectrum, which could be used for rogue wave detection.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 February 2015
- Quantization of β-Fermi–Pasta–Ulam lattice with nearest
and next-nearest neighbor interactions- Abstract: Publication date: 15 February 2015
Source:Physica D: Nonlinear Phenomena, Volume 294
Author(s): Aniruddha Kibey , Rupali Sonone , Bishwajyoti Dey , J. Chris Eilbeck
We quantize the β -Fermi–Pasta–Ulam (FPU) model with nearest and next-nearest neighbor interactions using a number conserving approximation and a numerically exact diagonalization method. Our numerical mean field bi-phonon spectrum shows excellent agreement with the analytic mean field results of Ivić and Tsironis (2006), except for the wave vector at the midpoint of the Brillouin zone. We then relax the mean field approximation and calculate the eigenvalue spectrum of the full Hamiltonian. We show the existence of multi-phonon bound states and analyze the properties of these states by varying the system parameters. From the calculation of the spatial correlation function we then show that these multi-phonon bound states are particle like states with finite spatial correlation. Accordingly we identify these multi-phonon bound states as the quantum equivalent of the breather solutions of the corresponding classical FPU model. The four-phonon spectrum of the system is then obtained and its properties are studied. We then generalize the study to an extended range interaction and consider the quantization of the β -FPU model with next-nearest-neighbor interactions. We analyze the effect of the next-nearest-neighbor interactions on the eigenvalue spectrum and the correlation functions of the system.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 February 2015
- Snakes and ladders in an inhomogeneous neural field model
- Abstract: Publication date: 15 February 2015
Source:Physica D: Nonlinear Phenomena, Volume 294
Author(s): Daniele Avitabile , Helmut Schmidt
Continuous neural field models with inhomogeneous synaptic connectivities are known to support traveling fronts as well as stable bumps of localized activity. We analyze stationary localized structures in a neural field model with periodic modulation of the synaptic connectivity kernel and find that they are arranged in a snakes-and-ladders bifurcation structure. In the case of Heaviside firing rates, we construct analytically symmetric and asymmetric states and hence derive closed-form expressions for the corresponding bifurcation diagrams. We show that the approach proposed by Beck and co-workers to analyze snaking solutions to the Swift–Hohenberg equation remains valid for the neural field model, even though the corresponding spatial–dynamical formulation is non-autonomous. We investigate how the modulation amplitude affects the bifurcation structure and compare numerical calculations for steep sigmoidal firing rates with analytic predictions valid in the Heaviside limit.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 February 2015
- Slowly varying control parameters, delayed bifurcations, and the stability
of spikes in reaction–diffusion systems- Abstract: Publication date: 1 January 2015
Source:Physica D: Nonlinear Phenomena, Volume 290
Author(s): J.C. Tzou , M.J. Ward , T. Kolokolnikov
We present three examples of delayed bifurcations for spike solutions of reaction–diffusion systems. The delay effect results as the system passes slowly from a stable to an unstable regime, and was previously analyzed in the context of ODE’s in Mandel and Erneux (1987). It was found that the instability would not be fully realized until the system had entered well into the unstable regime. The bifurcation is said to have been “delayed” relative to the threshold value computed directly from a linear stability analysis. In contrast to the study of Mandel and Erneux, we analyze the delay effect in systems of partial differential equations (PDE’s). In particular, for spike solutions of singularly perturbed generalized Gierer–Meinhardt and Gray–Scott models, we analyze three examples of delay resulting from slow passage into regimes of oscillatory and competition instability. In the first example, for the Gierer–Meinhardt model on the infinite real line, we analyze the delay resulting from slowly tuning a control parameter through a Hopf bifurcation. In the second example, we consider a Hopf bifurcation of the Gierer–Meinhardt model on a finite one-dimensional domain. In this scenario, as opposed to the extrinsic tuning of a system parameter through a bifurcation value, we analyze the delay of a bifurcation triggered by slow intrinsic dynamics of the PDE system. In the third example, we consider competition instabilities triggered by the extrinsic tuning of a feed rate parameter. In all three cases, we find that the system must pass well into the unstable regime before the onset of instability is fully observed, indicating delay. We also find that delay has an important effect on the eventual dynamics of the system in the unstable regime. We give analytic predictions for the magnitude of the delays as obtained through the analysis of certain explicitly solvable nonlocal eigenvalue problems (NLEP’s). The theory is confirmed by numerical solutions of the full PDE systems.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 1 January 2015
- Dynamics of elastic strands with rolling contact
- Abstract: Publication date: 15 February 2015
Source:Physica D: Nonlinear Phenomena, Volume 294
Author(s): François Gay-Balmaz , Vakhtang Putkaradze
We derive the equations of motion for rolling elastic strands in persistent rolling contact. The equations, presented first in an abstract form, are obtained by using the theory of Euler–Poincaré reduction by symmetries, appropriately modified to incorporate nonholonomic rolling conditions via the Lagrange–d’Alembert principle. We then show how to apply that theory to a particular case of elastic strands in rolling contact with naturally circular cross-section, when the deformation of the cross-section at contact is assumed to be negligible. We also derive a consistent geometric theory of rolling motion for discrete strands, or chains, in contact. The paper is concluded by showing highly non-trivial chaotic behavior even in the most simple configurations.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 February 2015
- Editorial Board
- Abstract: Publication date: 15 February 2015
Source:Physica D: Nonlinear Phenomena, Volume 294
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 February 2015
- On heteroclinic separators of magnetic fields in electrically conducting
fluids- Abstract: Publication date: 15 February 2015
Source:Physica D: Nonlinear Phenomena, Volume 294
Author(s): V. Grines , T. Medvedev , O. Pochinka , E. Zhuzhoma
In this paper we partly solve the problem of existence of separators of a magnetic field in plasma. We single out in plasma a 3-body with a boundary in which the movement of plasma is of special kind which we call an (a–d)-motion. We prove that if the body is the 3-annulus or the “fat” orientable surface with two holes then the magnetic field necessarily has a heteroclinic separator. The statement of the problem and the suggested method for its solution lead to some theoretical problems from Dynamical Systems Theory which are of interest of their own.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 February 2015
- A renormalization approach to the universality of scaling in phyllotaxis
- Abstract: Publication date: Available online 17 February 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Christian H. Reick
Phyllotaxis, i.e. the arrangement of plant organs like leaves, florets, scales, bracts etc. around a shoot, stem, or cone, is often highly regular. Across the plant kingdom phyllotaxis shows not only qualitatively, but also quantitatively identical features, like the occurence of divergence angles close to noble irrationals. In a previous study (Reick, 2012) a mechanism has been identified that explains the selection of these particular divergence angles on the basis of self-similarity and scaling, numerically found in the bifurcation diagrams of simple dynamical models of phyllataxis. In the present paper, by constructing a renormalization theory, the universality of this scaling is proved for a whole class of models, prototypically represented by Thornley’s model of phyllotaxis (Thornley, 1975). The renormalization is constructed from another self-similarity found numerically for the Fourier transform of the abstract potential governing the mutual inhibition of primordia. Surprisingly, the resulting renormalization transformation is already known from the treatment of the quasiperiodic transition to chaos but operates here on a different function space. It turns out that the fixed points of the renormalization transformation are characterized by divergences of the form Θ ( κ ) = 1 / τ ( κ ) , where, written as continued fraction, τ ( κ ) = [ κ ; κ , κ , … ] , κ ∈ N + . To show the universality of the scaling, it is demonstrated that the fixed points are unstable and that the associated scaling factors α ( κ ) = − ( τ ( κ ) ) 2 and β ( κ ) = τ ( κ ) are exactly those that were found in (Reick, 2012) to rule the selfsimilarity of the bifurcation structure. Thereby, the present paper puts forward an explanation for the universal appearance of certain phyllotactic patterns that is independent of physiological detail of plant growth.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: Available online 17 February 2015
- Dynamics and bifurcations in a Dn-symmetric Hamiltonian network.
Application to coupled gyroscopes- Abstract: Publication date: 1 January 2015
Source:Physica D: Nonlinear Phenomena, Volume 290
Author(s): Pietro-Luciano Buono , Bernard S. Chan , Antonio Palacios , Visarath In
The advent of novel engineered or smart materials, whose properties can be significantly altered in a controlled fashion by external stimuli, has stimulated the design and fabrication of smaller, faster, and more energy-efficient devices. As the need for even more powerful devices grows, networks have become popular alternatives to advance the fundamental limits of performance of individual units. In many cases, the collective rhythmic behavior of a network can be studied through the classical theory of nonlinear oscillators or through the more recent development of the coupled cell formalism. However, the current theory does not account yet for networks in which cells, or individual units, possess a Hamiltonian structure. One such example is a ring array of vibratory gyroscopes, where certain network topologies favor stable synchronized oscillations. Previous perturbation-based studies have shown that synchronized oscillations may, in principle, increase performance by reducing phase drift. The governing equations for larger array sizes are, however, not amenable to similar analysis. To circumvent this problem, the model equations are now reformulated in a Hamiltonian structure and the corresponding normal forms are derived. Through a normal form analysis, we investigate the effects of various coupling schemes and unravel the nature of the bifurcations that lead a ring of gyroscopes of any size into and out of synchronization. The Hamiltonian approach can, in principle, be readily extended to other symmetry-related systems.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 1 January 2015
- Editorial Board
- Abstract: Publication date: 1 January 2015
Source:Physica D: Nonlinear Phenomena, Volume 290
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 1 January 2015
- Stability and asymptotics in nematic liquid crystals under a small
Dirichlet data and a non-constant magnetic field- Abstract: Publication date: 1 January 2015
Source:Physica D: Nonlinear Phenomena, Volume 290
Author(s): Junichi Aramaki
We consider the stability of a critical point and asymptotics of the minimizers of the free energy functional with a small Dirichlet boundary data under a non-constant exterior magnetic field in nematic liquid crystals. In the author’s previous papers, we studied the stability of a critical point under the more general hypotheses but we had to assume the curl-free condition on the critical point for technical reason. However, in this paper, without the curl-free condition, we can get the similar result in the special case where the elastic coefficients are equal.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 1 January 2015
- Algebraic geometry and stability for integrable systems
- Abstract: Publication date: 15 January 2015
Source:Physica D: Nonlinear Phenomena, Volume 291
Author(s): Anton Izosimov
In 1970s, a method was developed for integration of nonlinear equations by means of algebraic geometry. Starting from a Lax representation with spectral parameter, the algebro-geometric method allows to solve the system explicitly in terms of theta functions of Riemann surfaces. However, the explicit formulas obtained in this way fail to answer qualitative questions such as whether a given singular solution is stable or not. In the present paper, the problem of stability for equilibrium points is considered, and it is shown that this problem can also be approached by means of algebraic geometry.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 January 2015
- Dimensionality reduction of collective motion by principal manifolds
- Abstract: Publication date: 15 January 2015
Source:Physica D: Nonlinear Phenomena, Volume 291
Author(s): Kelum Gajamannage , Sachit Butail , Maurizio Porfiri , Erik M. Bollt
While the existence of low-dimensional embedding manifolds has been shown in patterns of collective motion, the current battery of nonlinear dimensionality reduction methods is not amenable to the analysis of such manifolds. This is mainly due to the necessary spectral decomposition step, which limits control over the mapping from the original high-dimensional space to the embedding space. Here, we propose an alternative approach that demands a two-dimensional embedding which topologically summarizes the high-dimensional data. In this sense, our approach is closely related to the construction of one-dimensional principal curves that minimize orthogonal error to data points subject to smoothness constraints. Specifically, we construct a two-dimensional principal manifold directly in the high-dimensional space using cubic smoothing splines, and define the embedding coordinates in terms of geodesic distances. Thus, the mapping from the high-dimensional data to the manifold is defined in terms of local coordinates. Through representative examples, we show that compared to existing nonlinear dimensionality reduction methods, the principal manifold retains the original structure even in noisy and sparse datasets. The principal manifold finding algorithm is applied to configurations obtained from a dynamical system of multiple agents simulating a complex maneuver called predator mobbing, and the resulting two-dimensional embedding is compared with that of a well-established nonlinear dimensionality reduction method.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 January 2015
- Nonlinear mixed solitary—Shear waves and pulse equi-partition in a
granular network- Abstract: Publication date: 15 January 2015
Source:Physica D: Nonlinear Phenomena, Volume 291
Author(s): Yijing Zhang , M. Arif Hasan , Yuli Starosvetsky , D. Michael McFarland , Alexander F. Vakakis
We study primary pulse transmission in a two-dimensional granular network composed of two ordered chains that are nonlinearly coupled through Hertzian interactions. Impulsive excitation is applied to one of the chains (designated as ‘excited chain’), and the resulting transmitted primary pulses in both chains are considered, especially in the non-directly excited chain (the ‘absorbing chain’). A new type of mixed nonlinear solitary pulses–shear waves is predicted for this system, leading to primary pulse equi-partition between chains. An analytical reduced model for primary pulse transmission is derived to study the strongly nonlinear acoustics in the small-amplitude approximation. The model is re-scalable with energy and parameter-free, and is asymptotically solved by extending the one-dimensional nonlinear mapping technique of Starosvetsky (2012). The resulting nonlinear maps governing the amplitudes of the mixed-type waves accurately capture the primary pulse propagation in this system and predict the first occurrence of energy equipartition in the network. To confirm, in part, the theoretical results we experimentally test a series of two-dimensional granular networks, and prove the occurrence of strong energy exchanges leading to eventual pulse equi-partition between the excited and absorbing chains, provided that the number of beads is sufficiently large.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 January 2015
- Global attractors for quasilinear parabolic–hyperbolic equations
governing longitudinal motions of nonlinearly viscoelastic rods- Abstract: Publication date: 15 January 2015
Source:Physica D: Nonlinear Phenomena, Volume 291
Author(s): Stuart S. Antman , Süleyman Ulusoy
We prove the existence of a global attractor and estimate its dimension for a general family of third-order quasilinear parabolic–hyperbolic equations governing the longitudinal motion of nonlinearly viscoelastic rods carrying an end mass and subject to interesting body forces. The simplest version of the equations has the form w t t = n ( w x , w x t ) x where n is defined on ( 0 , ∞ ) × R and is a strictly increasing function of each of its arguments, with n → − ∞ as its first argument goes to 0. This limit characterizes a total compression, a source of technical difficulty, which new delicate a priori estimates prevent. We determine how the dimension of the attractor varies with the ratio of the mass of the rod to that of the end mass, giving conditions ensuring that the dimension is small. The estimates of dimension illuminate asymptotic analyses of the governing equation as this mass ratio goes to 0.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 January 2015
- Numerical bifurcation analysis of the bipedal spring-mass model
- Abstract: Publication date: 15 January 2015
Source:Physica D: Nonlinear Phenomena, Volume 291
Author(s): Andreas Merker , Dieter Kaiser , Martin Hermann
The spring–mass model and its numerous extensions are currently one of the best candidates for templates of human and animal locomotion. However, with increasing complexity, their applications can become very time-consuming. In this paper, we present an approach that is based on the calculation of bifurcations in the bipedal spring–mass model for walking. Since the bifurcations limit the region of stable walking, locomotion can be studied by computing the corresponding boundaries. Originally, the model was implemented as a hybrid dynamical system. Our new approach consists of the transformation of the series of initial value problems on different intervals into a single boundary value problem. Using this technique, discontinuities can be avoided and sophisticated numerical methods for studying parametrized nonlinear boundary value problems can be applied. Thus, appropriate extended systems are used to compute transcritical and period-doubling bifurcation points as well as turning points. We show that the resulting boundary value problems can be solved by the simple shooting method with sufficient accuracy, making the application of the more extensive multiple shooting superfluous. The proposed approach is fast, robust to numerical perturbations and allows determining complete manifolds of periodic solutions of the original problem.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 January 2015
- Riemann–Hilbert approach to gap probabilities for the Bessel process
- Abstract: Publication date: 1 March 2015
Source:Physica D: Nonlinear Phenomena, Volumes 295–296
Author(s): Manuela Girotti
We consider the gap probability for the Bessel process in the single-time and multi-time case. We prove that the scalar and matrix Fredholm determinants of such process can be expressed in terms of determinants of integrable kernels in the sense of Its–Izergin–Korepin–Slavnov and thus related to suitable Riemann–Hilbert problems. In the single-time case, we construct a Lax pair formalism and we derive a Painlevé III equation related to the Fredholm determinant.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 1 March 2015
- Core regulatory network motif underlies the ocellar complex patterning in
Drosophila melanogaster- Abstract: Publication date: 1 March 2015
Source:Physica D: Nonlinear Phenomena, Volumes 295–296
Author(s): D. Aguilar-Hidalgo , M.C. Lemos , A. Córdoba
During organogenesis, developmental programs governed by Gene Regulatory Networks (GRN) define the functionality, size and shape of the different constituents of living organisms. Robustness, thus, is an essential characteristic that GRNs need to fulfill in order to maintain viability and reproducibility in a species. In the present work we analyze the robustness of the patterning for the ocellar complex formation in Drosophila melanogaster fly. We have systematically pruned the GRN that drives the development of this visual system to obtain the minimum pathway able to satisfy this pattern. We found that the mechanism underlying the patterning obeys to the dynamics of a 3-nodes network motif with a double negative feedback loop fed by a morphogenetic gradient that triggers the inhibition in a French flag problem fashion. A Boolean modeling of the GRN confirms robustness in the patterning mechanism showing the same result for different network complexity levels. Interestingly, the network provides a steady state solution in the interocellar part of the patterning and an oscillatory regime in the ocelli. This theoretical result predicts that the ocellar pattern may underlie oscillatory dynamics in its genetic regulation.
Graphical abstract
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 1 March 2015
- The “hidden” dynamics of the Rössler attractor
- Abstract: Publication date: 1 March 2015
Source:Physica D: Nonlinear Phenomena, Volumes 295–296
Author(s): Dimitris T. Maris , Dimitris A. Goussis
The fast/slow dynamics of the Rössler model in the chaotic regime are compared to the dynamics of the reduced (slow) model governing the flow along the slow invariant manifold, on which the trajectory is restrained to evolve during the slow part of the attractor. It is shown that the dynamics of the reduced model incorporate the slow time scales of the full model. However, instead of the fast dissipative time scale of the full model that restrains the trajectory on the slow invariant manifold, the reduced model generates a new time scale that (i) relates to the curvature of the manifold and (ii) becomes faster as higher order corrections are incorporated in the approximation of the manifold and the reduced model. It is shown that this new time scale does not characterize the motion of the solution on the manifold but it can be employed as a signal for the strengthening or weakening of the manifold, depending on whether it relates to components of the reduced model that tend to lead its solution towards equilibrium or away from it. Finally, it is demonstrated that the new time scale introduced by the reduced model has a significant role in determining the maximum accuracy that can be delivered by the singular perturbation methodology.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 1 March 2015
- A numerical approach to blow-up issues for dispersive perturbations of
Burgers’ equation- Abstract: Publication date: 1 March 2015
Source:Physica D: Nonlinear Phenomena, Volumes 295–296
Author(s): Christian Klein , Jean-Claude Saut
We provide a detailed numerical study of various issues pertaining to the dynamics of the Burgers’ equation perturbed by a weak dispersive term: blow-up in finite time versus global existence, nature of the blow-up, existence for “long” times, and the decomposition of the initial data into solitary waves plus radiation. We numerically construct solitary waves for fractional Korteweg–de Vries equations.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 1 March 2015
- Modified inelastic bouncing ball model for describing the dynamics of
granular materials in a vibrated container- Abstract: Publication date: 15 January 2015
Source:Physica D: Nonlinear Phenomena, Volume 291
Author(s): Junius André F. Balista , Caesar Saloma
We show that at the onset of convection, the acceleration of a confined granular material is not necessarily equal to that of its vibrated container. Convection happens when the material is able to counter the downward gravitational pull and accelerates at a rate that is equal to the gravitational acceleration g . We modify the Inelastic Bouncing Ball Model and incorporate the transmissibility parameter T r which measures the efficiency that the external force driving the container is transmitted to the material itself. For a specified T r value, the material is represented by an inelastic bouncing ball with a time-of-flight T ( Γ ; T r ) where Γ = A 0 ω 2 / g , is the dimensionless container acceleration, and A 0 and ω are the driving amplitude and angular frequency, respectively. For a given Γ -range, the T ( Γ ; T r ) curve provides the bifurcation diagram of the perturbed material and a family of bifurcation diagrams is generated for a set of T r values. We illustrate that T r is useful in rationalizing experimental results produced by confined granular materials that is subjected to a range of applied force magnitudes. For the same physical set-up, the force transmission efficiency from the container to the grains may not remain constant as the force strength is varied. The efficiency is also affected by the presence or absence of air in the vibrated container.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 January 2015
- Stability of equilibrium state in a laser with rapidly oscillating delay
feedback- Abstract: Publication date: 15 January 2015
Source:Physica D: Nonlinear Phenomena, Volume 291
Author(s): E.V. Grigorieva , S.A. Kaschenko
Dynamics of laser with time-variable delayed feedback is analyzed in the neighborhood of the equilibrium. For the system, averaged over a rapidly variable, we obtain parameters at which the stationary state is stable. Stabilization of the stationary state due to modulation of the delay can be successful (unsuccessful) in domains adjacent to super (sub-) critical Hopf bifurcation boundaries. In a vicinity of the bifurcation points, stable and unstable periodic solutions are asymptotically described in dependence on the modulation frequency.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 January 2015
- Magnetisation oscillations by vortex–antivortex dipoles
- Abstract: Publication date: 15 January 2015
Source:Physica D: Nonlinear Phenomena, Volume 291
Author(s): Stavros Komineas
A vortex–antivortex dipole can be generated due to current with in-plane spin-polarisation, flowing into a magnetic element, which then behaves as a spin transfer oscillator. Its dynamics is analysed using the Landau–Lifshitz equation including a Slonczewski spin-torque term. We establish that the vortex dipole is set in steady state rotational motion due to the interaction between the vortices, while an external in-plane magnetic field can tune the frequency of rotation. The rotational motion is linked to the nonzero skyrmion number of the dipole. The spin-torque acts to stabilise the vortex dipole at a definite vortex–antivortex separation distance. In contrast to a free vortex dipole, the rotating pair under spin-polarised current is an attractor of the motion, therefore a stable state. The details of the rotating magnetisation configurations are analysed theoretically and numerically. The asymptotic behaviour of the rotating configurations provide results on their expected stability. Extensive numerical simulations reveal three types of vortex–antivortex pairs which are obtained as we vary the external field and spin-torque strength. We give a guide for the frequency of rotation based on analytical relations.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 January 2015
- Editorial Board
- Abstract: Publication date: 15 January 2015
Source:Physica D: Nonlinear Phenomena, Volume 291
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 January 2015
- (Non)Uniqueness of critical points in variational data assimilation
- Abstract: Publication date: Available online 24 January 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Graham Cox
In this paper we apply the 4D-Var data assimilation scheme to the initialization problem for a family of quasilinear evolution equations. The resulting variational problem is non-convex, so it need not have a unique minimizer. We comment on the implications of non-uniqueness for numerical applications, then prove uniqueness results in the following situations: 1) the observational times are sufficiently small; 2) the prior covariance is sufficiently small. We also give an example of a data set where the cost functional has a critical point of arbitrarily large Morse index, thus demonstrating that the geometry can be highly nonconvex even for a relatively mild nonlinearity.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: Available online 24 January 2015
- Instability observations associated with wave breaking in the
stable-stratified deep-ocean- Abstract: Publication date: 1 February 2015
Source:Physica D: Nonlinear Phenomena, Volumes 292–293
Author(s): Hans van Haren
High-resolution temperature observations above underwater topography in the deep, stably stratified ocean have revealed two distinctive turbulence processes. These processes are associated with different phases of a large-scale (here tidal) internal gravity wave: (i) highly nonlinear turbulent bores during the upslope propagating phase, and (ii) Kelvin–Helmholtz billows, at some distance above the slope, during the downslope phase. Whilst the former may be associated in part with convective turbulent overturning following Rayleigh–Taylor instabilities ‘RTi’, the latter is mainly related to shear-induced Kelvin–Helmholtz instabilities. In this paper, details are particularly presented of rare (convective) RTi penetrating stable density stratification under high-frequency internal waves. Such ‘apparent RTi’ can be explained using both stability parameterization of entrainment across a density interface, and, more relevant here, internal wave acceleration overcoming the reduced gravity.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 1 February 2015
- Mixed mode oscillations in a conceptual climate model
- Abstract: Publication date: 1 February 2015
Source:Physica D: Nonlinear Phenomena, Volumes 292–293
Author(s): Andrew Roberts , Esther Widiasih , Martin Wechselberger , Christopher K.R.T. Jones
Much work has been done on relaxation oscillations and other simple oscillators in conceptual climate models. However, the oscillatory patterns in climate data are often more complicated than what can be described by such mechanisms. This paper examines complex oscillatory behavior in climate data through the lens of mixed-mode oscillations. As a case study, a conceptual climate model with governing equations for global mean temperature, atmospheric carbon, and oceanic carbon is analyzed. The nondimensionalized model is a fast/slow system with one fast variable (corresponding to ice volume) and two slow variables (corresponding to the two carbon stores). Geometric singular perturbation theory is used to demonstrate the existence of a folded node singularity. A parameter regime is found in which (singular) trajectories that pass through the folded node are returned to the singular funnel in the limiting case where ϵ = 0 . In this parameter regime, the model has a stable periodic orbit of type 1 s for some s > 0 . To our knowledge, it is the first conceptual climate model demonstrated to have the capability to produce an MMO pattern.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 1 February 2015
- Premixed-flame shapes and polynomials
- Abstract: Publication date: 1 February 2015
Source:Physica D: Nonlinear Phenomena, Volumes 292–293
Author(s): Bruno Denet , Guy Joulin
The nonlinear nonlocal Michelson–Sivashinsky equation for isolated crests of unstable flames is studied, using pole-decompositions as starting point. Polynomials encoding the numerically computed 2 N flame-slope poles, and auxiliary ones, are found to closely follow a Meixner–Pollaczek recurrence; accurate steady crest shapes ensue for N ≥ 3 . Squeezed crests ruled by a discretized Burgers equation involve the same polynomials. Such explicit approximate shapes still lack for finite- N pole-decomposed periodic flames, despite another empirical recurrence.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 1 February 2015
- Coleman–Gurtin type equations with dynamic boundary conditions
- Abstract: Publication date: 1 February 2015
Source:Physica D: Nonlinear Phenomena, Volumes 292–293
Author(s): Ciprian G. Gal , Joseph L. Shomberg
We present a new formulation and generalization of the classical theory of heat conduction with or without fading memory. As a special case, we investigate the well-posedness of systems which consist of Coleman–Gurtin type equations subject to dynamic boundary conditions, also with memory. Nonlinear terms are defined on the interior of the domain and on the boundary and subject to either classical dissipation assumptions, or to a nonlinear balance condition in the sense of Gal (2012). Additionally, we do not assume that the interior and the boundary share the same memory kernel.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 1 February 2015
- Dynamics and stability of a discrete breather in a harmonically excited
chain with vibro-impact on-site potential- Abstract: Publication date: 1 February 2015
Source:Physica D: Nonlinear Phenomena, Volumes 292–293
Author(s): Nathan Perchikov , O.V. Gendelman
We investigate the existence and stability of discrete breathers in a chain of masses connected by linear springs and subjected to vibro-impact on-site potentials. The latter are comprised of harmonic springs and rigid constraints limiting the possible motion of the masses. Local dissipation is introduced through a non-unit restitution coefficient characterizing the impacts. The system is excited by uniform time-periodic forcing. The present work is aimed to study the existence and stability of similar breathers in the space of parameters, if additional harmonic potentials are introduced. Existence–stability patterns of the breathers in the parameter space and possible bifurcation scenarios are investigated analytically and numerically. In particular, it is shown that the addition of a harmonic on-site potential can substantially extend the stability domain, at least close to the anti-continuum limit. This result can be treated as an increase in the robustness of the breather from the perspective of possible practical applications.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 1 February 2015
- Editorial Board
- Abstract: Publication date: 1 February 2015
Source:Physica D: Nonlinear Phenomena, Volumes 292–293
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 1 February 2015
- Critical behavior for scalar nonlinear waves
- Abstract: Publication date: 1 February 2015
Source:Physica D: Nonlinear Phenomena, Volumes 292–293
Author(s): Davide Masoero , Andrea Raimondo , Pedro R.S. Antunes
In the long wave regime, nonlinear waves may undergo a phase transition from a smooth behavior to a fast oscillatory behavior. In this study, we consider this phenomenon, which is commonly known as dispersive shock, in the light of Dubrovin’s universality conjecture (Dubrovin, 2006; Dubrovin and Elaeva, 2012) and we argue that the transition can be described by a special solution of a model universal partial differential equation. This universal solution is constructed using the string equation. We provide a classification of universality classes and an explicit description of the transition with special functions, thereby extending Dubrovin’s universality conjecture to a wider class of equations. In particular, we show that the Benjamin–Ono equation belongs to a novel universality class with respect to those known previously, and we compute its string equation exactly. We describe our results using the language of statistical mechanics, where we show that dispersive shocks share many of the features of the tricritical point in statistical systems, and we also build a dictionary of the relations between nonlinear waves and statistical mechanics.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 1 February 2015
- Delay stabilizes stochastic motion of bumps in layered neural fields
- Abstract: Publication date: 1 March 2015
Source:Physica D: Nonlinear Phenomena, Volumes 295–296
Author(s): Zachary P. Kilpatrick
We study the effects of propagation delays on the stochastic dynamics of bumps in neural fields with multiple layers. In the absence of noise, each layer supports a stationary bump. Using linear stability analysis, we show that delayed coupling between layers causes translating perturbations of the bumps to decay in the noise-free system. Adding noise to the system causes bumps to wander as a random walk. However, coupling between layers can reduce the variability of this stochastic motion by canceling noise that perturbs bumps in opposite directions. Delays in interlaminar coupling can further reduce variability, since they couple bump positions to states from the past. We demonstrate these relationships by deriving an asymptotic approximation for the effective motion of bumps. This yields a stochastic delay-differential equation where each delayed term arises from an interlaminar coupling. The impact of delays is well approximated by using a small delay expansion, which allows us to compute the effective diffusion in bumps’ positions, accurately matching results from numerical simulations.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 1 March 2015
- Pitchfork–Hopf bifurcations in 1D neural field models with
transmission delays- Abstract: Publication date: 15 March 2015
Source:Physica D: Nonlinear Phenomena, Volume 297
Author(s): K. Dijkstra , S.A. van Gils , S.G. Janssens , Yu.A. Kuznetsov , S. Visser
Recently, local bifurcation theory for delayed neural fields was developed. In this paper, we show how symmetry arguments and residue calculus can be used to simplify the computation of the spectrum in special cases and the evaluation of the normal form coefficients, respectively. This is done hand in hand with an extensive study of two pitchfork–Hopf bifurcations for a 1D neural field model with ‘Wizard hat’ type connectivity.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 March 2015
- Barriers to transport and mixing in volume-preserving maps with nonzero
flux- Abstract: Publication date: 1 March 2015
Source:Physica D: Nonlinear Phenomena, Volumes 295–296
Author(s): Adam M. Fox , Rafael de la Llave
We identify some geometric structures (secondary tori) that restrict transport and prevent mixing in perturbations of integrable volume-preserving systems with nonzero net flux. Unlike the customary KAM tori, secondary tori cannot be continued to the tori present in the integrable system but are generated by resonances and have a contractible direction. We also note that secondary tori persist under the addition of a net flux, which destroys all customary KAM tori. We introduce a remarkably simple algorithm to analyze the behavior of volume preserving maps and to obtain quantitative properties of the secondary tori. We then implement the algorithm and, after running it, present assertions regarding the distribution of the escape times of the unbounded orbits, the abundance of secondary tori, the size of the resonant regions, and the robustness of the tori under the addition of a mean flux.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 1 March 2015
- Direct dynamical energy cascade in the modified KdV equation
- Abstract: Publication date: 15 March 2015
Source:Physica D: Nonlinear Phenomena, Volume 297
Author(s): Denys Dutykh , Elena Tobisch
In this study we examine the energy transfer mechanism during the nonlinear stage of the Modulational Instability (MI) in the modified Korteweg–de Vries (mKdV) equation. The particularity of this study consists in considering the problem essentially in the Fourier space. A dynamical energy cascade model of this process originally proposed for the focusing NLS-type equations is transposed to the mKdV setting using the existing connections between the KdV-type and NLS-type equations. The main predictions of the D -cascade model are outlined and validated by direct numerical simulations of the mKdV equation using the pseudo-spectral methods. The nonlinear stages of the MI evolution are also investigated for the mKdV equation.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 March 2015
- Complex short pulse and coupled complex short pulse equations
- Abstract: Publication date: 15 March 2015
Source:Physica D: Nonlinear Phenomena, Volume 297
Author(s): Bao-Feng Feng
In the present paper, we propose a complex short pulse equation and a coupled complex short equation to describe ultra-short pulse propagation in optical fibers. They are integrable due to the existence of Lax pairs and infinite number of conservation laws. Furthermore, we find their multi-soliton solutions in terms of pfaffians by virtue of Hirota’s bilinear method. One- and two-soliton solutions are investigated in details, showing favorable properties in modeling ultra-short pulses with a few optical cycles. Especially, same as the coupled nonlinear Schrödinger equation, there is an interesting phenomenon of energy redistribution in soliton interactions. It is expected that, for the ultra-short pulses, the complex and coupled complex short pulses equation will play the same roles as the nonlinear Schrödinger equation and coupled nonlinear Schrödinger equation.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 March 2015
- Long range annealing of defects in germanium by low energy plasma ions
- Abstract: Publication date: 15 March 2015
Source:Physica D: Nonlinear Phenomena, Volume 297
Author(s): J.F.R. Archilla , S.M.M. Coelho , F.D. Auret , V.I. Dubinko , V. Hizhnyakov
Ions arriving at a semiconductor surface with very low energy (2–8 eV) are interacting with defects deep inside the semiconductor. Several different defects were removed or modified in Sb-doped germanium, of which the E -center has the highest concentration. The low fluence and low energy of the plasma ions imply that the energy has to be able to travel in a localized way to be able to interact with defects up to a few microns below the semiconductor surface. After eliminating other possibilities (electric field, light, heat) we now conclude that moving intrinsic localized modes (ILMs), as a mechanism of long-distance energy transport, are the most likely cause. This would be striking evidence of the importance of ILMs in crystals and opens the way to further experiments to probe ILM properties both in semiconductors and in the metals used for contacts. Although most of the measurements have been performed on germanium, similar effects have been found in silicon.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 March 2015
- The long time behavior of Brownian motion in tilted periodic potentials
- Abstract: Publication date: 15 March 2015
Source:Physica D: Nonlinear Phenomena, Volume 297
Author(s): Liang Cheng , Nung Kwan Yip
A variety of phenomena in physics and other fields can be modeled as Brownian motion in a heat bath under tilted periodic potentials. We are interested in the long time average velocity considered as a function of the external force, that is, the tilt of the potential. In many cases, the long time behavior–pinning and de-pinning phenomenon–has been observed. We use the method of stochastic differential equation to study the Langevin equation describing such diffusion. In the over-damped limit, we show the convergence of the long time average velocity to that of the Smoluchowski–Kramers approximation, and carry out asymptotic analysis based on Risken’s and Reimann et al.’s formula. In the under-damped limit, applying Freidlin et al.’s theory, we first show the existence of three pinning and de-pinning thresholds of the normalized tilt, corresponding to the bi-stability phenomenon; and second, as noise approaches zero, derive formulas of the mean transition times between the pinning and running states.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 15 March 2015
- Arithmetic exponents in piecewise-affine planar maps
- Abstract: Publication date: 1 April 2015
Source:Physica D: Nonlinear Phenomena, Volumes 298–299
Author(s): John A.G. Roberts , Franco Vivaldi
We consider the growth of some indicators of arithmetical complexity of rational orbits of (piecewise) affine maps of the plane, with rational parameters. The exponential growth rates are expressed by a set of exponents; one exponent describes the growth rate of the so-called logarithmic height of the points of an orbit, while the others describe the growth rate of the size of such points, measured with respect to the p -adic metric. Here p is any prime number which divides the parameters of the map. We show that almost all the points in a domain of linearity (such as an elliptic island in an area-preserving map) have the same set of exponents. We also show that the convergence of the p -adic exponents may be non-uniform, with arbitrarily large fluctuations occurring arbitrarily close to any point. We explore numerically the behaviour of these quantities in the chaotic regions, in both area-preserving and dissipative systems. In the former case, we conjecture that wherever the Lyapunov exponent is zero, the arithmetical exponents achieve a local maximum.
PubDate: 2015-02-24T08:19:14Z
- Abstract: Publication date: 1 April 2015
- Phase description of oscillatory convection with a spatially translational
mode- Abstract: Publication date: Available online 26 December 2014
Source:Physica D: Nonlinear Phenomena
Author(s): Yoji Kawamura , Hiroya Nakao
We formulate a theory for the phase description of oscillatory convection in a cylindrical Hele-Shaw cell that is laterally periodic. This system possesses spatial translational symmetry in the lateral direction owing to the cylindrical shape as well as temporal translational symmetry. Oscillatory convection in this system is described by a limit-torus solution that possesses two phase modes; one is a spatial phase and the other is a temporal phase. The spatial and temporal phases indicate the “position” and “oscillation” of the convection, respectively. The theory developed in this paper can be considered as a phase reduction method for limit-torus solutions in infinite-dimensional dynamical systems, namely, limit-torus solutions to partial differential equations representing oscillatory convection with a spatially translational mode. We derive the phase sensitivity functions for spatial and temporal phases; these functions quantify the phase responses of the oscillatory convection to weak perturbations applied at each spatial point. Using the phase sensitivity functions, we characterize the spatiotemporal phase responses of oscillatory convection to weak spatial stimuli and analyze the spatiotemporal phase synchronization between weakly coupled systems of oscillatory convection.
PubDate: 2014-12-30T06:53:56Z
- Abstract: Publication date: Available online 26 December 2014
- Data-driven non-Markovian closure models
- Abstract: Publication date: Available online 23 December 2014
Source:Physica D: Nonlinear Phenomena
Author(s): Dmitri Kondrashov , Mickaël D. Chekroun , Michael Ghil
This paper has two interrelated foci: (i) obtaining stable and efficient data-driven closure models by using a multivariate time series of partial observations from a large-dimensional system; and (ii) comparing these closure models with the optimal closures predicted by the Mori-Zwanzig (MZ) formalism of statistical physics. Multilayer stochastic models (MSMs) are introduced as both a very broad generalization and a time-continuos limit of existing multilevel, regression-based approaches to closure in a data-driven setting; these approaches include empirical model reduction (EMR), as well as more recent multi-layer modeling. It is shown that the multilayer structure of MSMs can provide a natural Markov approximation to the generalized Langevin equation (GLE) of the MZ formalism. A simple correlation-based stopping criterion for an EMR-MSM model is derived to assess how well it approximates the GLE solution. Sufficient conditions are derived on the structure of the nonlinear cross-interactions between the constitutive layers of a given MSM to guarantee the existence of a global random attractor. This existence ensures that no blow-up can occur for a very broad class of MSM applications, a class that includes non-polynomial predictors and nonlinearities that do not necessarily preserve quadratic energy invariants. The EMR-MSM methodology is applied to a conceptual, nonlinear, stochastic climate model of coupled slow and fast variables, in which only slow variables are observed. It is shown that the resulting closure model with energy-conserving nonlinearities efficiently captures the main statistical features of the slow variables, even when there is no formal scale separation and the fast variables are quite energetic. Second, an MSM is shown to successfully reproduce the statistics of a partially observed, generalized Lokta-Volterra model of population dynamics in its chaotic regime. The challenges here include the rarity of strange attractors in the model’s parameter space and the existence of multiple attractor basins with fractal boundaries. The positivity constraint on the solutions’ components replaces here the quadratic-energy–preserving constraint of fluid-flow problems and it successfully prevents blow-up.
PubDate: 2014-12-25T15:33:57Z
- Abstract: Publication date: Available online 23 December 2014
- Convergence of the 2D Euler-α to Euler equations in the Dirichlet
case: Indifference to boundary layers- Abstract: Publication date: 1 February 2015
Source:Physica D: Nonlinear Phenomena, Volumes 292–293
Author(s): Milton C. Lopes Filho , Helena J. Nussenzveig Lopes , Edriss S. Titi , Aibin Zang
In this article we consider the Euler- α system as a regularization of the incompressible Euler equations in a smooth, two-dimensional, bounded domain. For the limiting Euler system we consider the usual non-penetration boundary condition, while, for the Euler- α regularization, we use velocity vanishing at the boundary. We also assume that the initial velocities for the Euler- α system approximate, in a suitable sense, as the regularization parameter α → 0 , the initial velocity for the limiting Euler system. For small values of α , this situation leads to a boundary layer, which is the main concern of this work. Our main result is that, under appropriate regularity assumptions, and despite the presence of this boundary layer, the solutions of the Euler- α system converge, as α → 0 , to the corresponding solution of the Euler equations, in L 2 in space, uniformly in time. We also present an example involving parallel flows, in order to illustrate the indifference to the boundary layer of the α → 0 limit, which underlies our work.
PubDate: 2014-12-16T17:32:44Z
- Abstract: Publication date: 1 February 2015